I don't think poklitics would change the nature of the language very much (though depensding on the greater scheme of things, you could see more or less High German, French, Spanish and English loan words and different standardisation). Rather, I think that politics could change the designation of the language. Norwegian is a separate language from danish today because of politics. The Norwegians would probably still speak much as they do if it was thought of as Danish, at least until the impact of universal schooling and cheap press.
The dialects of Low German spoken along the upper Rhine and the North Sea coast were closer to Dutch then they were to the Meißner High German that would displace them. If they were instead in a political system dominated from the Netherlands, there is no reason to think their speakers would adopt High German. Political realities suggest, though, that the cultural ands economic entre of this area would be in Holland. IOTL the Netherlands drew large numbers of workers from these lands well into the 19th century. Dutch fleets and armies were filled with natives of Dithmarschen, the East Frisian Islands, Westfalia and the Elbe estuary. They adapted quickly to Dutch. In a different setting, they could easily make it their status dialect. In this world, Reformed church literature from Leyden instead of Lutheran from Erfurt and Wittenberg, the books of Amserdam printshops instad of the Leipzig and Frankfurt fairs will guide the style of the leading classes. They can continue to consider their language Low German. It is not far from düdesch to duits.